Convex solutions of certain elliptic equations have constant rank Hessians
type: Item
instance of: scholarly article
title: Convex solutions of certain elliptic equations have constant rank Hessians
zbMATH Open document ID: 0624.35031
DOI: 10.1007/BF00279844
author: Nicholas J. Korevaar
author: John L. Lewis
published in: Archive for Rational Mechanics and Analysis
publication date: 1987-01-01T00:00:00Z
review text: In recent years there have been several results on the convexity of solutions to certain elliptic equations, e.g. \textit{N. Korevaar} [Indiana Univ. Math. J. 32, 603-614 (1983; Zbl 0481.35024)], \textit{A. Kennington} [ibid. 34, 687-704 (1985; Zbl 0549.35025)] and the reviewer [Z. Angew. Math. Mech. 64, No.5, T 364-T 366 (1984; Zbl 0581.73006); Math. Methods Appl. Sci. 8, 93-101 (1986; Zbl 0616.35006); Commun. Partial Differ. Equations 10, 1213-1225 (1985; Zbl 0587.35026)]. In contrast to these results the authors of the paper under review assume that u is convex and solves \(\Delta u=f(u,\nabla u)>0\) in \(\Omega \subset {\mathbb{R}}^ n\). The main result is the following. If 1/f(\(\cdot,\nabla u)\) is convex in u, then the Hessian H of u has constant rank in \(\Omega\). This implies in particular, that H has full rank in \(\Omega\) if it has full rank in a single point of \(\Omega\). The proof is a nontrivial extension of a former result of \textit{L. Caffarelli} and \textit{A. Friedman} [Duke Math. J. 52, 431-456 (1985; Zbl 0599.35065)]. The paper contains also applications of the main result.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 4013141
zbMATH Keywords: semilinear
zbMATH Keywords: convexity of solutions
zbMATH Keywords: Hessian
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4013141